Your fret board looks to about as thin is you can get without having it come apart. The vintage Martins have the frets go clear through to the neck. Must have used a piece by piece to attach them.

I believe that they used cut the slots part way, glue the board to the neck then cut the rest of the way by hand...Or cut the slots on the neck with a set up like I have, I can saw the fret slots either on or off the neck...I’ll try and show some pics of the set up.